Abstract
In pointed braided fusion categories knowing the self-symmetry braiding of simples is theoretically enough to reconstruct the associator and braiding on the entire category (up to twisting by a braided monoidal auto-equivalence). We address the problem to provide explicit associator formulas given only such input. This problem was solved by Quinn in the case of finitely many simples. We reprove and generalize this in various ways. In particular, we show that extra symmetries of Quinn’s associator can still be arranged to hold in situations where one has infinitely many isoclasses of simples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baues, H.: Combinatorial homotopy and 4-dimensional complexes, De Gruyter Expositions in Mathematics, vol. 2. Walter de Gruyter & Co., Berlin (1991). With a preface by Ronald Brown. MR 1096295
Bulacu, D., Caenepeel, S., Torrecillas, B.: The braided monoidal structures on the category of vector spaces graded by the Klein group, Proc. Edinb. Math. Soc. (2) 54(3), 613–641. MR 2837470 (2011)
Braunling, O.: Braided categorical groups and strictifying associators. Homol. Homotopy Appl. 22(2), 295–321 MR 4098945 (2020)
Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories. I, Selecta Math. (N.S.) 16(1), 1–119. MR 2609644 (2010)
Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories, Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence (2015). MR 3242743
Eilenberg, S., Mac Lane, S.: On the groups H(π,n). I, Ann. of Math. (2) 58, 55–106. MR 0056295 (1953)
Flanders, H.: Tensor and exterior powers. J. Algebra 7, 1–24. MR 212044 (1967)
Huang, H.-L., Liu, G., Ye, Y.: The braided monoidal structures on a class of linear Gr-categories. Algebr. Represent. Theory 17(4), 1249–1265. MR 3228486 (2014)
Huang, H.-L., Liu, G., Yang, Y., Ye, Y.: Finite quasi-quantum groups of diagonal type, J. Reine Angew. Math. 759, 201–243. MR 4058179 (2020)
Huang, H.-L, Wan, Z, Ye, Y: Explicit cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf-Witten invariants. Proc. Roy. Soc. Edinburgh Sect. A 150(4), 1937–1964. MR 4122441 (2020)
Johnson, N., Osorno, A.: Modeling stable one-types, Theory Appl. Categ. 26(20), 520–537. MR 2981952 (2012)
Joyal, A., Street, R.: Braided monoidal categories. Macquarie Mathematical Reports, no 860081 (1986)
Joyal, A.: Braided tensor categories. Adv. Math. 102(1), 20–78. MR 1250465 (1993)
Kapustin, A., Saulina, N.: Topological boundary conditions in abelian Chern-Simons theory. Nuclear Phys. B 845(3), 393–435. MR 2755172 (2011)
Mac Lane, S.: Cohomology theory of Abelian groups. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., vol. 2, Amer. Math. Soc., Providence, R. I., 1952, pp. 8–14. MR 0045115 (1950)
Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323. Springer, Berlin. MR 2392026 (2008)
Quinn, F.: Group categories and their field theories. Proceedings of the Kirbyfest, Berkeley. Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 407–453. MR 1734419 (1998)
Sính, H.X.: Gr-catégories (thesis, handwritten manuscript). Université, Paris 7. https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html (1975)
Whitehead, J.H.C.: A certain exact sequencex. Ann. Math. (2) 52, 51–110. MR 35997 (1950)
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Alistair Savage
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author was supported by DFG GK1821 “Cohomological Methods in Geometry”. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Braunling, O. Quinn’s Formula and Abelian 3-Cocycles for Quadratic Forms. Algebr Represent Theor 24, 1523–1555 (2021). https://doi.org/10.1007/s10468-020-10001-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-020-10001-1